bricks in the wall


much of the point here is to “explain why”
the figure in the upper right… “pascal’s triangle”
works the way it does: students gladly learn
in minutes how to write out rows by adding
pairs taken from previous rows (according to
the formula in the lower left, though they
know it not; eventually they’ll have to be
encouraged to care but of course one need
not speak of thorny notational issues at
every single opportunity; when the student
is ready the code’ll have been there waiting
all along). the thing is: what the heck does
this ritual-adding-together have to do
with “counting subsets”, then, eh? because
when we “use” these numbers in our little
probability-and-counting problems back
in class, we’ll be referring to “N choose R”
all the time: the number of ways to
“choose” a set of R things from a set of N
.
well, put like that…
the thing is more or less obviously
then to look carefully at arrangements-of-subsets
and see if we can piece out
“what comes from where”.

my best shot so far.
better with the handwaving of course.

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  1. Due to its simple construction by factorials, a very basic representation of Pascal’s triangle in terms of the matrix exponential can be given: Pascal’s triangle is the exponential of the matrix which has the sequence 1, 2, 3, 4, … on its subdiagonal and zero everywhere else.

    good ol’ wikipedia.

  2. okay suppose you’ve got to know about N
    but all you know about is N-1. somebody
    gives you N things anyhow and sez
    “how many ways can i get R things
    out of these here?”.

    take one of the N things and paint it blue.

    now there’s the N-1 colorless things and the blue one.
    we’re supposed to pick a set of R things somehow.
    okay… i say now there are *two* ways i might
    go about getting such a set:
    pick the *whole set of R things*
    from the nonblue N-1…
    there are {N-1 \choose R} ways to do this…
    or
    pick *only R-1 nonblue things* from
    this same N-1, plus the blue thing for
    the full quota of R things altogether:
    there are {N-1 \choose R-1} ways
    to do this.

    so we’ll’ve seen that
    {N \choose R} = {N-1 \choose R} + {N-1 \choose R-1}.
    (not the same formula as in
    my handwritten display but
    the same fact.)

    i forget who put it to me like this
    but this is the one that stuck.

  3. vlorbik

    back on the message of the day.
    i did a new issue of the
    “lectures without words”
    series today: MathEdZine 0.8
    (the K_n — K_4 remix).
    it’s a classic “minicomic” format:
    copy on both sides of a sheet
    of typing paper, fold the long way
    first, then fold again the short way,
    staple along the spine (with an
    ordinary desk stapler this will
    entail folding over a few pages;
    with a “longarm” stapler one
    can just shove it it and bang down),
    and trim it at the top. voila:
    an eight-pager just like cynicalman.
    without the brilliant art and
    longpracticed production values.
    anyhow, it’s the first zine i’ve made
    with this particular “imposition” and
    so i get a tiny bit of a thrill from it.
    there’s also a sort of, oh-heck-give-up
    to the whole thing though since i’d
    never even thought of *doing* a remix
    if the “unstaplebook” format of
    the original 0.6 and 0.7 wasn’t
    distracting almost every subject
    i’ve had a chance to work with…

    i’m glad i *did* the unstaplebooks.
    part of the point *is* the very
    what-the-heck-is-*this*
    effect that the very format
    of the document inspires…
    among other things, this gives the “teacher”
    (that would be me in the interviews i’ve had
    but is always somebody else in my imagination)
    a chance to bail out on the math
    (i’ve-never-been-good-in-math
    is just about as reliable a response
    to being confronted with a MEdZ
    as what-the-heck-is-this is) and
    talk about *publishing* and, more
    to the point for “educators” i suppose,
    *self*-publishing and the DIY vibe
    generally. teach a kid to write up
    cool math facts and they’ll fish
    for the rest of their life kinda thing.

    but i’m laying off of ’em
    (“unstaplebook” micro- and nano-
    -zines) once i’ve filled in the
    \Bbb R and \Bbb C
    issues at 0.4 and 0.5
    i think.

  4. vlorbik

    the display in the photo
    appears in both K_n
    and the “remix” issue.

    i meant to’ve said.
    (why doesn’t somebody
    just *interrupt* me
    when i get going like that.)

  5. I’m going to try to respond to kate(t)’s challenge by using binomial expansion to create pascal’s triangle. Maybe.

    1
    x | y
    first put xs on the left: xx xy, then put ys on the left: yx yy

    xx | xy yx | yy

    first put xs on the left: xxx xxy xyx xyy, then put ys on the left: yxx yxy yyx yyy

    xxx | xxy xyx yxx | xyy yxy yyx | yyy

    counting, we get 1 2 1 0 + 0 1 2 1

    like, no surprise, multiplication by 11.

    Need to play and clean, and develop cooler graphics than I’ve ever done.

    Jonathan

  6. it looks to me like “pascal’s triangle”
    … why do the binomial coefficients
    behave the way they do…
    is the heart of the matter for
    kate’s problem. everything else
    is easier for *me* to see, anyhow.

    i’ve *also* got notes on how to
    lay out answers and work ’em
    by rote. i drilled several classes
    in expanding (P + Q)^N
    for probability-and-stix
    applications for quite a bit
    more than the standard presentations
    would ever dream of doing just
    so i’d feel like they’d learned
    to do *something* calling for
    honest algebraic manipulation
    of binomial coefficients.
    i had to cut out some of the
    “pretend to learn a bunch of
    technical terms for regurgitation”
    stuff to do it though. oh the horror.

    i’m unlikely to submit anything myself.
    i’ve finally given in to the need to
    develop cooler graphics than i’ve ever done.
    with ink and paper and whatnot though…
    making real objects in the real world.
    computers are just way too expensive
    and frustrating without massive
    institutional support.

    i can’t work this damn copier and
    it breaks my heart.




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